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\topic{Lecture 1 \\Differential Calculus-I\\ \scriptsize Successive Differentiation (15 Sep 2009)}
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If $y$ be a function of $x$, its differential cofficient $\frac{dy}{dx}$ will be a function of $x$, which can be differentiated further. The next differential cofficient of $\frac{dy}{dx}$ is called the second differential coefficient of $y$. Similarly, the differential cofficient of the second differential coefficient is called the third differential coefficient of $y$, and so on. The successive differential coefficients of $y$ with respect to $x$ are denoted as
\[
\frac{dy}{dx},~\frac{d^2y}{dx^2},~\frac{d^3y}{dx^3},~...
\]
Thus, the $n$th differential coefficient of $y$ is denoted as $\frac{d^ny}{dx^n}$.\footnote{Some alternative notations for $n$th derivative of $y$ with respect to $x$ are - $\frac{d^n}{dx^n}y$,  $d^ny/dx^n$, $D^ny$, $y_n$, $y^{(n)}$, etc.}
\section{Standard Results}
\begin{enumerate}
	\item $D^n(ax+b)^m = m(m-1)(m-2) ... (m-n+1)a^n(ax+b)^{m-n}$
	\item $D^n(ax+b)^{-1} = (-1)^n n!a^n(ax+b)^{-n-1}$
	\item $D^ne^{ax} = a^ne^{ax}$
	\item $D^na^{x} = (\log a)^na^{x}$
	\item $D^n \log(ax+b) = \frac{(-1)^{n-1} (n-1)!a^n}{(ax+b)^n}$
	\item $D^n \sin(ax+b) = a^{n} \sin(ax+b+n \pi/2)$
	\item $D^n \cos(ax+b) = a^{n} \cos(ax+b+n \pi/2)$
	\item $D^n {e^{ax}\sin(bx+c)} = r^{n}e^{ax} \sin(bx+c+n \phi)$, where $r=\sqrt{a^2+b^2}$, and $\phi = \tan^{-1} \frac{b}{a}$
	\item $D^n e^{ax}\cos(bx+c) = r^{n}e^{ax} \cos(bx+c+n \phi)$, where $r=\sqrt{a^2+b^2}$, and $\phi = \tan^{-1} \frac{b}{a}$
\end{enumerate}
\section{Computation of $n$th Derivative}
These results may be used directely to compute $n$th derivative of some given function of $x$. Some times it becomes necessary to use the tools like partial fraction, De-Moivre's Theorem, Trigonometrical Transformations to reduce the given function in appropriate form and then we apply these standard results. Look up some examples here.
\begin{example}
Find the $n$th differential coefficient of $\frac{x^3}{(x-1)(x-2)}$
\end{example}

We know by algebra of partial fraction that
\[\frac{x^3}{(x-1)(x-2)} = x+3 + \frac{7x-6}{(x-1)(x-2)} =x+3 - \frac{1}{(x-1)}+ \frac{8}{(x-2)}\]
Therefore, if $n > 1$,
\[\frac{d^n}{dx^n}\frac{x^3}{(x-1)(x-2)} = (-1)^{n+1}n! \left[\frac{1}{(x-1)^{n+1}}- \frac{8}{(x-2)^{n+1}}\right]\]
\begin{example}
Find the $n$th differential coefficient of $\sin^5x \cos^3x$
\end{example}
Let 
\[ \cos x + i \sin x =z; ~\textrm{then} ~\cos x - i \sin x =v\]
Therefore
\[ 2\cos x =z+z^{-1},~~ 2i\sin x =z-z^{-1}\]
Also by using De Moivre's Theorem
\[ 2\cos px =z^p+z^{-p},~~ 2i\sin x =z^p-z^{-p}\]
Therefore 
\[2^52^3i^5 \sin^5x \cos^3x = (z-z^{-1})^5(z+z^{-1})^3 \]
\[=(z^8-z^{-8})-2(z^6-z^{-6})-2(z^4-z^{-4})+6(z^2-z^{-2})\]
\[=2i \sin 8x - 4i \sin 6x -4i \sin 4x + 12i \sin 2x\]
Now use standard results to evaluate required result
\[D^n(\sin^5x \cos^3x) \]
\[= 2^{-7}[8^n \sin (8x+n\pi /2) - 2.6^n \sin (6x+n\pi /2) -2.4^n \sin (4x+n\pi /2) + 6.2^n \sin (2x+n\pi /2)]\]
Hence.

\section*{Problems}
\begin{enumerate}
\item  Find $\frac{d^{3} y}{dx^{3} } $, when $y = x^3 + 4x + 2$.

\item   If $y = \sin(a\sin^{-1}x)$, prove that \[(1- x^2)\frac{d^2y}{dx^2}=x \frac{dy}{dx} - a^2y \]

\item   If $y = a \cos (\log x) + b \sin(\log x)$, prove that $x^2y'' + xy'+ y = 0$.

\item   If $p^2 = a^2 \cos 2\theta+ b^2 \sin 2\theta$, prove that $p + \frac{d^{2} p}{d \theta ^{2} } =\frac{a^{2} b^{2} }{p^{3} } $.

\item   If x = a (cos$\theta $+$\theta \sin \theta $), y =$a\left(\sin \theta -\theta \cos \theta \right)$, prove that$a\theta \frac{d^{2} y}{dx^{2} } =\sec ^{3} \theta $.

\item   Find the $n$th differential co-efficient of $\log (ax + x^2)$

\item   If $y = \cos 3x$, find $y_n$. 

\item   If $y = \sin2x \sin3x$, find $y_n$. 

\item   Find the $n$th differential co-efficient of $e^x  \sin 3x$.

\item   If $y =\frac{1}{1-5x+6x^{2} }$, find $y_n$.

\item   Find $n$th derivative of $\frac{ax+b}{cx+d}$.

\item   If $y =\sqrt{x+a}$, find $y_n$.

\item   Find nth derivative of $\frac{1}{x^{2} -a^{2} }$.

\item   If $y = \frac{x^{2} }{(x-1)^{2} (x+2)} $ , find $n$th derivative of $y$.

\item   Find $n$th derivative of $\tan ^{-1} \left(\frac{2x}{1-x^{2} } \right)$. 

\item   Prove that the value of the $n$th differential co-efficient of $\frac{x^{3} }{x^{2} -1} $ for $x = 0$ is zero, if $n$ is even; and is $-n!$ if $n$ is odd and greater than $1$.

\item   If $y = x \log \frac{x-1}{x+1} $, show that $y_n = (-1)^{n-2}(n-2)! \left[\frac{x-n}{(x-1)^{n}}-\frac{x+n}{(x+1)^{n} } \right]$.

\item   Find $n$th derivative of $x^2 \sin x$. 

\item   Find $n$th derivative of $e^x \log x$.

\item   If $y = x^2 e^x$, show that $\frac{d^{n} y}{dx^{n} } =\frac{1}{2} n(n-1)\frac{dy}{dx} +\frac{1}{2} (n-1)(n-2)y$.

\item   Differentiate $n$ times the following equations: 
				(a)  $(1- x^2)\frac{d^2y}{dx^2}-x \frac{dy}{dx} + a^{2} y =0$ (b)b $x^2 y_2 + xy_1+ y = 0$

\item   If $y = \sin (m \sin^{-1}x)$, then prove that $(1- x^2) y_2 - x y_1 + m^2y = 0$.
\end{enumerate}


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